It is important to distinguish between pure mathematics and its applications. This distinction is pertinent to the distinction between probability and density.
Probability is the fractional concentration of an element in a purely logical set. Density is the fractional concentration of a physical element in a physical set. Let the probability of blue marbles in a set of logical marbles be 0.5. For a comparable physical set of marbles the density of blue marbles is 1 blue marble per 2 marbles. In contrast to the probability, the density of blue marbles is not 0.5, but 0.5 blue marbles per marble.
The necessity of such a distinction is seen in the following example. If we form (even logical) packages of 2 marbles each on the basis of a density of 0.5 blue marbles per marble, all packages will have exactly one and only one blue marble per package. If we form logical packages of two marbles each on the basis of a probability of blue marbles of 0.5, we will have defined a logical population of packages in which the proportions of packages with 0, 1 and 2 blue marbles are 1:2:1.
Note: the formation of material sets on the basis of the probabilities of a source set can only be analogous to or a simulation of the purely logical concepts of mathematical probability.
Probability is purely a logical concept. The formation of new sets based on the probabilities of a source set is called random selection from the source set. This is strictly logical. No physical thing can be selected with a total disregard to its physical properties, yet this is what is required by the concept of random selection. Thus the formation of new physical sets based on the probabilities of a physical source set, can only be an analogy of the mathematics. In such analogies, human ignorance of the IDs during selection is equated with randomness.
In contrast to probability, which is solely logical, density can be characteristic of material reality. An example is the density of carbon atoms in glucose, which is one carbon atom per four atoms.
An example of confusing density and probability is the calculation of the number of earthlike planets in the universe by Richard Dawkins in The God Delusion, page 138. He conservatively estimates the number of planets in the universe as one billion, billion. Assuming a density of 1 earthlike planet per billion planets, he correctly calculates the number of earthlike planets in the universe as 1 billion.
That, however, is not how Dawkins describes his calculation. Instead of assuming a density of 1 earthlike planet per billion planets, Dawkins claims he is assuming a staggeringly, absurdly, stupefyingly low probability of earthlike planets of 1 per billion. If Dawkins were assuming a probability rather than a density, he would not have been calculating the number of earthlike planets in the universe. He would have been defining a population of universes, which would cover the spectrum from 0 earthlike planets per universe to 1 billion, billion earthlike planets per universe. In the defined population, modal universes would contain exactly I billion earthlike planets. However, this mode would be a very small percentage of the total population of universes. This very small percentage would be the probability that a randomly selected universe would contain exactly 1 billion planets.
In my next post I will present a more subtle and intriguing instance of conflating probability and density.